Inference of Cultural Transmission Modes Based on Incomplete Information

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1 Wayne State University Human Biology Open Access Pre-Prints WSU Press Inference of Cultural Transmission Modes Based on Incomplete Information Bryan Wilder University of Central Florida, Anne Kandler City University London, Recommended Citation Wilder, Bryan and Kandler, Anne, "Inference of Cultural Transmission Modes Based on Incomplete Information" (2015). Human Biology Open Access Pre-Prints This Article is brought to you for free and open access by the WSU Press at It has been accepted for inclusion in Human Biology Open Access Pre-Prints by an authorized administrator of

2 Inference of Cultural Transmission Modes Based on Incomplete Information Bryan Wilder 1 and Anne Kandler 2 * 1 University of Central Florida. 2 City Universoty London. *Correspondence to: Anne Kandler, City University London. anne.kandler.1@city.ac.uk. KEY WORDS: CULTURAL TRANSMISSION, STATISTICAL INFERENCE, SAMPLING SCHEME. Short title: Inference of Cultural Transmission Modes Abstract In this paper we explore the theoretical limits of the inference of cultural transmission modes based on sparse population-level data. We approach this problem by investigating whether different transmission modes produce different temporal dynamics of cultural change. In particular we explore whether the distributions of the average time a variant stays the most common variant in the population, denoted by t max, conditioned on the considered transmission modes are sufficiently different to allow for inference of underlying transmission modes. We assume time series data detailing the frequencies of different variants of a cultural trait in a population at different points in time and investigate the temporal

3 resolution (i.e. the length of the time series and the distance between consecutive time points) that is needed to ensure distinguishability between transmission modes. We find that under complete information most transmission modes can be distinguished on the base of the statistic t max, however we should not expect the same results if only infrequent information about the most common cultural variant in the population are available. The question of how ideas, knowledge or even behaviours are transmitted between different individuals of a population has sparked an enormous amount of theoretical and empirical research in the last decade (e.g., Baum et al. 2004; Bentley et al. 2004; Henrich 2001; Mesoudi and O'Brien 2008; Rendell et al. 2010; Rendell et al. 2011). Naturally fine-grained individual-level data detailing who learns from whom would be most suited to answer these questions empirically. However outside experimental conditions, large individual data sets of this kind are difficult to obtain, especially in pre-modern contexts. Studies such as Benheim et al. (2014) who used historical records of the board game GO, or Henrich and Broesch (2011) who used contemporary data from Fiji to determine the relative importance of different cultural transmission processes are remarkable exceptions. Often population-level data describing the frequencies of different cultural variants in a population at different points in time are the only information researchers have about the process of cultural transmission (Shennan 2011). But the assumption that temporal changes in the frequencies of cultural variants over time reflect the dynamic of the underlying cultural transmission process (Boyd and Richerson 1985; Cavalli-Sforza

4 and Feldman 1981) leads naturally to the question of how accurate those transmission modes can be inferred from potentially sparse, population-level data. Recently researchers have started exploring the effects of time-averaging on the accuracy of identifying cultural transmission modes, in particular neutral evolution, based on archaeological assemblages (Madsen 2012; Premo 2014; Porčić 2014). Time-averaging is defined as a phenomenon whereby cultural remains deposited at different times come to be preserved together and thus appear to be penecontemporaneous (Premo 2014; Kidwell and Behrensmeyer 1993; Kowalewski 1996) and it has been shown that it diminishes our ability to correctly detect patterns of neutral evolution in population-level data. Further, Eerkens et al. (2005) investigated the relationship between phylogenetic signature in population-level data and cultural transmission modes and concluded that some transmission modes maintain phylogenetic signature better than others. In this paper we contribute to the discussion of how well cultural transmission modes can be inferred from population-level data by exploring the theoretical limits of such a statistical inference procedure. We base our considerations on time series data detailing the frequencies of different variants of a cultural trait in a population at different points in time and approach the inference problem by investigating whether different transmission modes result in different values of certain population-level statistics. If this is the case then transmission modes are distinguishable and obtained estimates of those statistics are informative about the underlying transmission mode. Conversely, if transmission modes are not distinguishable (or only partially distinguishable) then different transmission modes may result in similar values of

5 the population-level statistics and consequently could have produced obtained estimates (Cavalli-Sforza and Feldman 1981). In Kandler et al. (submitted) we developed a simulation framework which tracks the temporal changes in frequency of different variants of a cultural trait under the assumptions that cultural transmission occurs in a vertical, horizontal, oblique or unbiased way. Thereby i. vertical transmission describes the transmission from parents to offspring, ii. horizontal transmission describes the transmission between (related or unrelated) individuals of the same age group, iii. oblique transmission describes the transmission from individuals of one age group to individuals of younger age groups who is not their direct descendant and iv. unbiased transmission describes the transmission between all individuals of the population (see, e.g., Cavalli-Sforza and Feldman 1981). Importantly, the framework produces frequency data, similar to observable ones, conditioned on the considered transmission modes. This enables us to ask the following question: Given the generated frequency data, how well can we theoretically infer the (known) underlying cultural transmission process? We found that the frequency distribution of the cultural variants in the population at a single point in time does not carry a

6 strong enough signature about the transmission process. That is, we cannot reliably distinguish between transmission modes on the basis of, e.g., the level of cultural diversity in the population. However, the temporal dynamic of cultural change possesses more information. In situations of high rates of cultural transmission we can distinguish between vertical and horizontal transmission, vertical and unbiased transmission, unbiased and oblique transmission and horizontal and oblique transmission on the base of the average time the most common variant remains the most frequent in the population. But crucially, this result relies on the fact that we have information about the most common variant at every point in time which is, especially for historic case studies, very rarely the case. So what is the minimum temporal resolution of the data that is needed to guarantee these distinguishability results? We note that cultural transmission can occur in many more ways than the four transmission modes considered here (see, e.g., Boyd and Richerson 1985; Laland 2004; Morgan et al. 2012), but the aim of the study is to explore the data resolution that is needed to obtain distinguishability results close to the theoretically possible results. In the following we briefly summarise the simulation framework and the statistical inference procedure developed in Kandler et al. (submitted). In section 2 we then assume that we only have information about the most frequent variant at a certain number of points in time and analyse how incomplete (temporal) information influences the power of the inference procedure. This also allows us to give suggestions about the minimum time resolution of the sample that is needed to guarantee theoretical distinguishability between the different transmission modes.

7 The Model We consider a population of N individuals where each individual is characterised by a set of four discrete cultural traits. Those traits can take five different variants and we assume that the traits differ in the way they are transmitted. In detail, the variables v describe traits which are transmitted vertically, h describe traits which are initially transmitted vertically and horizontally during an individual's life time, o describe traits which are initially transmitted vertically and obliquely during an individual's life time, u describe traits which are initially transmitted vertically and in an unbiased way during an individual's life time. We further assume that an offspring inherits all cultural traits from its parent but may have the opportunity for cultural transmission during its lifetime. At each point in time an individual is described by the variables I with x v, h, o, u summarizing individual i's acquired variants. It holds I 1,,5 for x = v, h, o, u. Further, we define five age groups covering the ages 0 10; 11 20; 21 30; 31 40; 41 50, respectively and the variable I "# denotes individual i's age group. Now at each time step (which is equal to 10 years) all individuals are subject to a number of

8 demographic and cultural processes (as detailed below) resulting in frequency changes of the five variants of the cultural traits in the population. Demographic Processes At each time step individuals age and have a chance of reproducing and dying. Aging means that individuals move to the next age group. We assume the maximum age to be 50 and consequently all individuals moving to age group six die with probability 1. Additionally, individuals can die with probability p "#$% in each time step. We further assume a constant population size equal to the carrying capacity (denoted by N). Therefore the number of individuals entering the population is given by the difference between N and the number of deaths in the considered time step. Reproduction occurs asexually; an individual of the age groups 2 or 3 is chosen at random and produces an offspring which inherits its trait values. However, with a small probability μ a mutation occurs and the offspring inherits a trait value different from the parent's. This process of reproduction is repeated until the carrying capacity N is reached. Together these demographic processes produce a pyramid-shaped age structure with fewer individuals in older generations. It is possible that all potential parents die before reproducing, resulting in the extinction of the population. The results shown in this paper, however, are based on simulation runs where the population survived for the entire duration of the simulation.

9 Cultural Processes An individual acquires and changes its cultural phenotype through the process of cultural transmission. In the case of vertical transmission individual i inherits the cultural variant from its parent with probability 1 μ whereas with probability μ a mutation occurs and individual i acquires a variant randomly selected from the remaining four alternatives. Vertically transmitted traits do not change during an individual s lifetime. We stress that all traits are initially transmitted vertically, however in the case of horizontal, oblique and random transmission an individual engages with a probability p "#$"% in interactions with other individuals of the population in each time step and potentially change the values of the traits that are transmitted in a horizontal, oblique and unbiased manner. In detail, if transmission occurs horizontally, individual i randomly chooses another individual j from its own age group and with probability 1 μ it holds I = I with I "# = I "#. (1) If transmission occurs obliquely, individual i randomly chooses another individual j from an older age group and with probability 1 μ it holds I = I with I "# < I "#. (2)

10 If transmission occurs randomly, individual i randomly chooses another individual j from the entire population regardless of its age group and with probability 1 μ it holds I = I. (3) In all transmission events mutations occur with probability μ. In this case an individual adopts a randomly selected variant. Consequently, together with the interaction probability p "#$"% the mutation rate μ controls the rate of change in the population. It holds: The higher p "#$"% and μ, the more the cultural composition of the population can change in every time step. The simulation framework tracks the frequencies of the different variants of the considered cultural traits in the population conditioned on a specific mode of transmission at every point in time. Using this framework we explored in (Kandler et al. submitted) the distinguishability of transmission modes based on population-level frequency data. Importantly we found that the temporal dynamic of cultural change carries a stronger signature of the underlying transmission modes than the knowledge of the whole cultural composition of the population a specific point in time. To describe the temporal dynamic of cultural change we determined the average time a variant stays the most common variant in the population, denoted by t "#. This statistic is a measure for quantifying the speed of cultural change. Bentley et al. (2007) (and further analysed by Evans and Giometto 2011) described the temporal dynamic using turnover rates (defined as the number of new variants that

11 enter the list of the l variants with the highest frequency in the population). However, due to our restriction to five possible cultural variants turnover rates cannot be meaningfully applied to our situation. Additionally, the t "# statistic requires only knowledge of the cultural variant with the maximum frequency which often can be determined with higher certainty than the complete frequency distribution of the cultural variants present in the population. To explore the effects of time averaging we follow the approach suggested by Premo (2014). Using the framework described above we determine (timeaveraged) populations of cultural variants at each time point t by randomly sampling N variants from a pool of variants generated by the non-averaged populations at the time points t-i, i=1,,a. The effects of time-averaging with different strengths can be investigated by varying the parameter a. Statistical Inference Our framework generates, among others, the variant with the highest frequency in averaged and non-averaged populations at every time point t, t=1,,200 conditioned on the considered transmission modes. Based on this information we can estimate the average time t "# a variant stays the most common variant in the population (see section 2). To quantify the level of distinguishability of the considered modes based on an empirical estimate of t "# we calculate the overlap of the probability distributions of t "# for the transmission modes x = v, h, o, u. In more detail, the distributions of t "# conditioned on a transmission mode x indicate the average time a variant stays the most common variant in a population generated

12 by x. Therefore the size of the area of overlap, denoted by O ", between two distributions (see Figure for an example) produced by the transmission modes x and y, respectively, reflects the certainty with which the corresponding transmission modes can be distinguished based on an estimate of t "#. Figure 1. Illustration of the area of overlap O xy.. If O " is sufficiently close to zero then the transmission modes result mostly in different values of and consequently distinguishability can be guaranteed (cf. (Kandler et al. submitted) for details on the distinguishability criterion). On the contrary, a value close to one indicates that the probability distributions are almost identical and therefore the transmission modes cannot be distinguished on the base of t "#. In (Kandler et al. submitted) we have shown that for high rates of change (e.g. for p "#$"% = 1 which corresponds to on average five transmission events during an individual's life time and μ = 0.1) we can distinguish between vertical and horizontal transmission, vertical and unbiased transmission, unbiased and oblique transmission and horizontal and oblique transmission based on the statistic t "#. In this situation the areas of overlap O " between the probability distributions

13 produced by the different transmission modes are almost 0. Reducing the rate of change (e.g. reducing p "#$"% and/or μ) results in larger areas of overlap O " and consequently less certainty in distinguishing between transmission modes. However, this conclusion was reached under the assumption that we have full knowledge about the most common variant at every point in the considered time interval. This however, might only rarely be the case in historic case studies. Therefore we explore in the following the temporal resolution of the time series that is needed in order to obtain inference results close to what is theoretically possible. Distinguishability of Cultural Transmission Modes on the Base of Incomplete Information In the following we assume that we have knowledge about the most common variant at certain points in time but have no information about the cultural composition of the population for the remaining time points (see Figure Error Reference source not found. for an illustration). Varying the number of sample points and the distance between the sample points then allows us to determine the minimum requirements to still be able to distinguish between different transmission modes on the base of an estimate of t "#. We focus only on the case where there is a high rate of chance in the population. Otherwise, different transmission modes cannot be distinguished, even with perfect information (Kandler et al. submitted).

14 Figure 2. Illustration of the available information. The most common variant in the population (represented by different colours) is known for only a subset of the time points in the considered interval. Estimation of t "# We start our analysis by defining how the average time a variant stays the most common variant, t "#, is estimated based on incomplete temporal information about the cultural composition of the population. We consider the time interval [1,...,200], and the time series v(t),,"" records the variant with maximum frequency in the population at each time step t. It holds v t 1,,5. Now a sample of size n with n 200 is taken at the time points t = t,, t 1,,200 and the observations S = v(t ) describe the available information about the temporal dynamic of cultural change. Further, the distances between the sample points are denoted by = t t, i = 1,, n 1. Based on the sample S we can determine the time points t,, t t where the most common variant has changed. The variable n denotes the number of changes observed in the sample and the average time a variant stays the most common in the population can then be estimated by t "# = t + N t + t t (n 1). (4)

15 Naturally the accuracy of the estimate t "# is affected by the properties of the sample S in particular by its size n and the distances,, between the sample points. To illustrate these influences we consider the time interval [1,,200] and determine the distributions of the estimates t "# for horizontal transmission and sample sizes n=5;10;20;30;40 (Figure a). The distance between the sample points is chosen to be = = 3, i.e. every three time points we obtain information about the most common variant in the population. The dashed line in Figure a shows the distribution of t "# based on the whole time series and we observe: The larger n, the tighter is the distribution of t "#. However, the `true' distribution of t "# is not well approximated even for relatively large n pointing to the crucial importance of the distances = between sample points. Figure b shows the distributions of the estimates t "# for horizontal transmission and = = 1; 3; 5; 7; 10. The sample size is assumed to be n=20. Unsurprisingly, only for = 1 the `true' distribution of t "# can be well approximated. In this situation an increase of the sample size n results in a more accurate approximation of t "#. a. b. Figure 3. Distributions of the estimates t max for horizontal transmission and a) n=5;10;20;30;40$ and i = = 3 and b) n=20 and i = = 1; 3; 5; 7; 10.

16 In the following we explore for which values of n and we can distinguish between vertical and horizontal transmission, vertical and unbiased transmission, unbiased and oblique transmission and horizontal and oblique transmission and therefore obtain the theoretically possible distinguishability on the basis of incomplete information. Hypothesis Test Generally, to explore whether an estimate t "# derived from the empirical observations S = v(t ) at the time points t = t,, t is consistent with vertical, horizontal, oblique or unbiased transmission we need to determine the distributions P(t "# t ) of the average time a variant stays the most common variant in the population for each transmission mode x v, h, o, u. To do so we use the simulation framework and generate a time series v (t),, conditioned on the transmission mode x with x, h, o, u. We then determine the theoretical sample S = v (t ) based on the given sampling scheme t = t,, t and use equation (4) to calculate the estimate of t "#. Repeating this process allows us to determine the distributions P(t "# t ) of t "# conditioned on the transmission mode x v, h, o, u and the available sample points t = t,, t. Now if the estimate t "# is outside the 1 α 100% prediction interval of P(t "# t ) then we conclude that the transmission mode x could not have produced the observed data. Conversely, if the estimate falls within the prediction interval we cannot rule out transmission mode x as possible underlying evolutionary force. But how much information about the underlying transmission modes can be revealed by such a test? Even though we know that for t = [1,, N] we can

17 perfectly distinguish between vertical and horizontal transmission, vertical and unbiased transmission, unbiased and oblique transmission and horizontal and oblique transmission on the base of an estimate of t "#, how much is of this power is lost when considering only a sample at the time points t? To explore these questions we start by analysing the size of the type II error of the suggested hypothesis test for different sampling schemes t (n, ) (The type II error indicates the probability of failing to reject an incorrect null hypothesis.). We again generate data v (t),..., conditioned on the transmission mode y v, h, o, u, determine the sample S based on t (n, ) and calculate the estimates t "# according to equation (4). Lastly we compare the estimate t "# with the 1 α 100% confidence interval of the distribution P(t "# t ) of t "# under transmission mode x, denoted by a, b, and record whether the hypothesis H: Data was generated under transmission mode x is rejected. Repeating this process gives us the probability p " (t ) of rejecting transmission mode x when the data was produced by transmission mode y. Consequently 1 p " (t ) describes the type II error and it holds p " t = 1 f "# z t dz = 1 F "# b t + F "# (a t ) where f "# t describes the density function of the statistic t "# under transmission mode y and sample points t. F "# t denotes the corresponding distribution function.

18 Additionally, the knowledge of the theoretical distributions P(t "# t ) and P(t "# t ) allow us to quantify their area of overlap O " (t ) as a measure of the distinguishability under incomplete information. If the area of overlap is small then we can conclude that on the base of observations at the time points t (n, ) the transmission modes x and y are distinguishable. Results In the following we analyse the rejection probability p " (t ) and the area of overlap O " (t ) with x, y v, h, o, u for varying sample sizes n and distances between the time points and explore which sequence of sample points t = t (n, ) allows for reliable inference results. <B>Dependence on Sample Size n.</b> Figure shows the rejection probability p " (t ) as a function of the sample size n and = = 3. Thereby the observations v (t ) are generated by vertical (Figure a), horizontal (Figure b), oblique (Figure c) and unbiased transmission (Figure d) and the solid lines illustrate the rejection probabilities p " (t ) with x v, h, o, u, x y. The dashed lines describe the maximum rejection probability that can be obtained when observing every time point, i.e. t = [1,,200].

19 a. b. c. d. Figure 4. Rejection probability p xy (t S ) with x, y v, h, o, u as a function of n and i = = 3, N=100, μ = 0. 1, p within = 1 and data generated according to a) vertical transmission, b) horizontal transmission, c) oblique transmission and d) unbiased transmission. The red lines illustrate the rejection probability of the hypothesis of vertical transmission, the green lines of horizontal transmission, the blue lines of oblique transmission and the magenta lines of unbiased transmission. The corresponding dashed lines show the highest possible rejection probability determined from observations at all time points [1,,200]. Unsurprisingly for small sample sizes n the probability of rejecting the wrong transmission mode is very small. The estimate t "# is not precise enough to capture the differences in the temporal dynamic of cultural change. However, also for

20 increased n we only obtain rejection probabilities higher than 80% for vertical and horizontal transmission and vertical and unbiased transmission implying that only these transmission modes can be distinguished reliably on the base of incomplete information about the process of cultural change over the time interval [0,200]. Consequently even though it is theoretically possibly to distinguish between horizontal and oblique transmission and unbiased and oblique transmission we should not expect the same results when only incomplete information is available. Figure a shows the area of overlap O " (t ) of the distributions P(t "# t ) and P(t "# t ) produced by the different transmission modes for n=5,10;20;30;40;50;60 and = = 3. The squares show the difference from horizontal (green), oblique (blue) and unbiased (magenta) transmission to vertical transmission and therefore quantify the effect of cultural transmission occurring during the life time of an individual on the level of cultural diversity. The triangles show the difference from oblique (blue) and horizontal (green) transmission to unbiased transmission and therefore quantify the effect of restricting the set of possible transmission partners to partners within the same age group and to older partners, respectively. Lastly, the circles show the difference between horizontal transmission and oblique transmission. In agreement with Figure we observe that the area of overlap O " (t ) decreases for increasing n, however only vertical and horizontal transmission and vertical and unbiased transmission result in an area of overlap sufficiently small to ensure distinguishability. Figure b shows the areas of overlap O " (t ) for = = 1 and demonstrates that the ability to distinguish between transmission modes crucially depend on the chosen distance between the

21 sample points,. Interestingly for the majority of combinations of x and y with x, y v, h, o, u we obtain worse distinguishability results for =1 than for = 3. We note that analyses with = = 10 produced worse distinguishability results throughout. This points to the fact that there exists an optimal distance between the observations, which we will investigate in the next section. a. b. x Figure 5. Area of overlap O xy (t S ) between the distribution P(t max t S ) and y P(t max t S ) for x, y v, h, o, u and a) i = = 3 and b) i = = 1. The squares show the difference between horizontal (green), oblique (blue) and random (magenta) transmission to vertical transmission, the triangles show the difference between oblique (blue) and horizontal (green) transmission to unbiased transmission, and the circles show the difference between horizontal transmission to oblique transmission. Dependence on i. We have seen in the last section that variations in the distance between the sample points t and t greatly influence the distinguishability results. In the

22 following, we present the results of the analysis based on constant distance between the sample points, i.e. = but note that we carried out the same analysis with exponentially distributed distances, i.e. t t exp ( ) for Δ 2, and obtained very similar results (cf. Figure ). Figure shows the rejection probability p " (t ) with x, y v, h, o, u in dependence of. The sample size is assumed to be n=20. Similarly to Figure observations v (t ) are generated by vertical (Figure a), horizontal (Figure b), oblique (Figure c) and unbiased transmission (Figure d) and the solid lines illustrate the rejection probabilities p " (t ) with x v, h, o, u, x y. The dashed lines describe the maximum rejection probability that can be obtained when observing every time point, i.e. t = [1,,200]. We firstly observe that for each pair of transmission modes (x,y) there exists a which produces the maximum rejection probability p " (t ). As already indicated by Figure those maximum rejection probabilities are often not obtained for = 1. Intermediate values of ranging from 2 to 5 provide the best distinguishability results for the assumed situations of cultural change (N=100, μ = 0.1, p "#$"% = 1) under most transmission modes. Consequently, the sampling scheme that provides the best inference result is the one that produces distributions P(t "# t ) and P(t "# t ) as different as possible and not necessarily the most accurate estimation of t "#. Further we observe that the rejection probabilities p " (t ) decrease for large values of indicating that samples taken at too distant time points do not carry much information about the underlying process; the time series becomes more and more uncorrelated.

23 a. b. c. d. Figure 6. Rejection probability p xy (t S ) with x, y {v, h, o, u} as a function of i = for N=100, μ = 0. 1, p within = 1 and data generated according to a) vertical transmission, b) horizontal transmission, c) oblique transmission and d) unbiased transmission. The red lines illustrate the rejection probability of the hypothesis of vertical transmission, the green lines of horizontal transmission, the blue lines of oblique transmission and the magenta lines of unbiased transmission. The corresponding dashed lines show the highest possible rejection probability determined from observations at all time points [1,,200]. It is assumed n= 20. Figure shows the area of overlap O " (t ) of the distributions P(t "# t ) and P(t "# t ) produced by the different transmission modes for = = 1; 2; 4; 6; 8; 10 and n=20. We observe a similar behaviour: almost every

24 combination of x and y with x, y {v, h, o, u} requires a different value of to achieve the smallest area of overlap. a. b. x Figure 7. Area of overlap O xy (t S ) between the distributions P(t max t S ) and y P(t max t S ) for x, y {v, h, o, u} and n=20. Figure a: i = = 1; 2; 4; 6; 8; 10 and Figure b: t i1 t i exp ( ) for Δ 2. Colours and shapes correspond to the one in Figure. Effects of Time-Averaging To explore the effects of time-averaging we determine (time-averaged) populations of cultural variants at each time point t by randomly sampling N variants from a pool of variants generated by the non-averaged populations at the time points t-i, i=1,,a. Thus, the time series v "#$%# (t) now records the variant with,,"" maximum frequency in the averaged population whereby averaging occurred over the last a time steps. Figure shows the results of the distinguishability analysis for = = 1; 2; 4; 6; 8; 10, n=20 and averaging windows of size a=5 (Figure a) and a=10 (Figure b). It is obvious that the areas of overlap O (t "#$%#, ) of the distributions t "#$%#, ) and P(t "# t "#$%#, ) produced by the different transmission P(t "#

25 modes are increased compared to the situation without time-averaging (cf. Figure ). It holds: the larger the averaging window a the larger the area of overlap and therefore the smaller the certainty with which two transmission modes can be distinguished. However, even for a=10 the probability distributions t "#$%#, ) and P(t "# t "#$%#, ) are not identical. In the best situation we P(t "# obtain an overlap of roughly 45% indicating that either smaller or larger values of t "# can only be reached by one transmission mode or the other. x Figure 8. Area of overlap O xy (t S ) between the distributions P(t max t average,s ) and y P(t max t average,s ) for x, y {v, h, o, u}, n=20, a) a=5 and b) a=10. Colours and shape correspond to the one in Figure. To summarize, time-averaging dilutes the detectable signal of cultural transmission modes in population-level data but does not remove it completely. Conclusion In this paper we used a simulation framework to explore whether vertical, horizontal, oblique and unbiased transmission can be distinguished on the basis of

26 incomplete information about temporal change in the cultural composition of the population. We built our analysis on insights gained in a previous study (Kandler et al. submitted) where we showed that the temporal dynamic of cultural change carries a stronger signature of the underlying transmission modes than the knowledge of the whole cultural composition of the population a specific point in time expressed by, e.g., the level of cultural diversity. To describe the temporal dynamic of cultural change we determined the average time a variant stays the most common variant in the population, denoted by t "#. Using the simulation framework we generated the distributions (t "# ), {v, h, o, u} of t "# conditioned on the considered transmission modes x. These distributions illustrate the values of t "# that are possible under x and therefore the overlap between two distributions P(t "# ) and P(t "# ) provides a measure of the certainty with which two transmission modes x and y can be distinguished. Only if these two distributions are sufficiently different can distinguishability can be guaranteed. We have shown in (Kandler et al. submitted) that for high rates of change (i.e., individuals have frequent opportunities to engage in cultural transmission during their lifetime) the temporal dynamic of cultural change is very different between vertical and horizontal transmission, vertical and oblique transmission, vertical and unbiased transmission, unbiased and oblique transmission and horizontal and oblique transmission and concluded that it is possible to perfectly distinguish between these transmission modes on the base of the statistic t "#. Reducing the rate of change resulted in probability distributions with a greater overlap and consequently less certainty in distinguishing between transmission modes based on an estimate of

27 t "#. Crucially, this conclusion was reached under the assumption that we have full knowledge about the most common variant at every point in the considered time interval. This however, might only rarely be the case, especially in historic case studies. Therefore we explored in this paper whether the obtained distinguishability results hold for situations with incomplete temporal information about the most common variant in the population. We assumed that samples are taken at the time points t = {t,, t } and investigated the influence of the sample size n and the distance between the sample points on the ability to infer underlying transmission modes from population-level data. The average time a variant stays the most common variant in the population is determined by equation (4). Naturally those estimates are only precise if sufficiently many observations at consecutive time steps (i.e. = 1) are available. However, the aim of the analysis is not to obtain the most precise estimate of t "# but to determine the sampling scheme t = t (n, { }) that produces distributions of t "# which are as different as possible under the considered transmission modes. For that n should be chosen as high as possible. But as transmission modes usually operate on different time scales we found that different pairs (x,y) of transmission modes require different values of to achieve the smallest possible area of overlap O " (t ) for a given n (see Figure ). Consequently we need to determine the value of which provides an acceptable statistical power to distinguish between all considered transmission modes. We suggest that simulation frameworks, similar to the one considered here, which capture the main dynamics of the analysed case study could help determining the properties of the sampling scheme that produce

28 the highest possible power to distinguish between transmission modes. In situations where data has been already collected at time points t a simulation framework can explore whether it is theoretically possible to distinguish between transmission modes on the base of a single estimate of t "#. Additionally, the simulation framework generates for each sampling scheme and transmission mode x the distribution P(t "# t ) of t "# allowing for statistical hypothesis testing. In summary, even though it is possible to perfectly distinguish between vertical and horizontal transmission, vertical and unbiased transmission, unbiased and oblique transmission and horizontal and oblique transmission on the base of complete information about the most common variant in the population we should not expect the same result if only a sample taken at the time points t is available (see e.g. Figure and Figure ). In situations of incomplete information the level of distinguishability depends crucially on the properties of the sample. We have shown that the distance between the time points at which the samples are taken has a strong influence on amount of information about the underlying transmission mode can be inferred from population-level data. If is too large, i.e. the sample points are too distant from each other, then an increase in the sample size n will not improve the results greatly. If the observed population-level data is a product of time-averaging processes we should expect even less certainty in distinguishing between different cultural transmission modes. However, the detectable signal of transmission modes in population-level data but is not removed completely. Consequently it is unlikely that sparse population-level data such as knowledge about the most common cultural variant in the population at certain

29 points in time will allow for unique inference of the cultural transmission mode that produced the data. Different transmission modes will be consistent with the data. However, this conclusion does not render mathematical modelling meaningless. To the contrary, it is firstly important to understand the theoretical limits to statistical inference procedures and therefore to understand which kind of questions can be answered with which kind of data. Secondly the analysis of population level data might help exclude transmission modes that could not have produced the observed data and in this way lead to a reduction of the pool of potential hypotheses. Subsequently, different lines of evidence might be used to reduce this pool even further. <ACK><C>Acknowledgments</C> The authors would like to thank Tanmoy Bhattacharya for insightful discussions on this topic. Further, we thank three anonymous reviewers for their constructive comments which helped improving the paper. This research was supported by an NSF EAGER grant Linking Pattern and Process in Cultural Evolution. </ACK> Received 31 December 2014; revision accepted for publication 19 June Literature Cited Baum, W. M., P. J. Richerson, C. M. Efferson et al Cultural evolution in laboratory microsocieties including traditions of rule giving and rule following. Evol. Hum. Behav. 25:

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